Posets for which Verdier duality holds

We discuss two known sheaf-cosheaf duality theorems: Curry's for the face posets of finite regular CW complexes and Lurie's for compact Hausdorff spaces, i.e., covariant Verdier duality. We provide a uniform formulation for them and prove their generalizations. Our version of the former works over the sphere spectrum and for more general finite posets, which we characterize in terms of the Gorenstein* condition. Our version of the latter says that the stabilization of a proper separated $\infty$-topos is rigid in the sense of Gaitsgory. As an application, for stratified topological spaces, we clarify the relation between these two duality equivalences.